Subgroups — C₂²×#16810449 #27537113

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	2	yes	maximal
2	C₂×#16810449	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
5	C₂×#13133151	—	2	6	no	—
6	C₂×#13133151	C₂²	1	12	yes	—
7	#16810449	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₄	1	8	yes	complemented
10	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
11	C₂³×Dic₅₃	—	4	2	no	—
12	C₂×C₄⋊Dic₅₃	—	2	12	no	—
13	C₂×C₄⋊Dic₅₃	—	4	6	no	—
14	#13133151	—	2	12	no	—
15	#13133151	C₂³	1	8	yes	—
16	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₃	D₄	1	3	yes	—
19	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	24	yes	complemented
22	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
23	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
25	C₂²×C₄×C₅₃	D₈	1	3	yes	—
26	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
28	C₂²×Dic₅₃	—	4	12	no	—
29	C₂²×Dic₅₃	—	8	8	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
32	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
33	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
34	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	2	8	no	—
36	C₄⋊Dic₅₃	—	4	12	no	—
37	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
38	C₂³×C₅₃	#107	1	4	yes	—
39	C₂³×C₅₃	D₁₆	1	4	yes	—
40	C₂³×C₅₃	Q₃₂	1	4	yes	—
41	C₂³×C₅₃	C₂×D₈	1	3	yes	—
42	C₂×C₄×C₅₃	#107	1	12	yes	—
43	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
44	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
45	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
46	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	4	8	no	—
48	C₂×Dic₅₃	—	8	24	no	—
49	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
50	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×#366	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	#366	1	16	yes	complemented
54	C₂²×C₅₃	C₂×#107	1	6	yes	—
55	C₂²×C₅₃	C₂×D₁₆	1	6	yes	—
56	C₂²×C₅₃	C₂×Q₃₂	1	6	yes	—
57	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×#107	1	6	yes	—
59	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
61	Dic₅₃	—	8	8	no	—
62	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
63	C₂²×#107	—	53	2	no	—
64	C₂×#366	—	53	12	no	—
65	C₂×C₅₃	C₂×#366	1	12	yes	complemented
66	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
67	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
68	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
69	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
70	C₂²×C₄⋊C₄	—	106	2	no	—
71	C₂²×C₁₆	Dic₅₃	1	8	yes	complemented
72	C₂²×C₁₆	C₂×D₅₃	1	6	yes	—
73	C₂×#107	—	53	12	no	—
74	C₂×#107	—	106	6	no	—
75	#366	—	53	16	no	—
76	C₅₃	C₂²×#366	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₄	—	212	2	no	—
78	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
79	C₂²×C₈	Dic₁₀₆	1	4	yes	—
80	C₂²×C₈	D₂₁₂	1	3	yes	—
81	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
82	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
83	C₂×C₄⋊C₄	—	106	12	no	—
84	C₂×C₄⋊C₄	—	212	6	no	—
85	C₂×C₁₆	C₂×Dic₅₃	1	24	yes	complemented
86	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
87	#107	—	53	8	no	—
88	#107	—	106	12	no	—
89	C₂⁴	D₄₂₄	1	1	yes	characteristic
90	C₂²×C₄	—	424	8	no	—
91	C₂²×C₄	—	212	12	no	—
92	C₂²×C₄	D₄₂₄	1	3	yes	—
93	C₂²×C₄	Dic₂₁₂	1	4	yes	—
94	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
95	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
96	C₂×C₈	C₄⋊Dic₅₃	1	12	yes	—
97	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
98	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
99	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
100	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
101	C₄⋊C₄	—	106	8	no	—
102	C₄⋊C₄	—	212	12	no	—
103	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
104	C₂³	#13133151	1	4	yes	—
105	C₂³	D₈₄₈	1	4	yes	—
106	C₂³	Dic₄₂₄	1	4	yes	—
107	C₂³	C₂×D₄₂₄	1	3	yes	—
108	C₂×C₄	—	424	24	no	—
109	C₂×C₄	—	212	8	no	—
110	C₂×C₄	#13133151	1	12	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
112	C₂×C₄	C₂×D₄₂₄	1	3	yes	—
113	C₂×C₄	C₂×Dic₂₁₂	1	6	yes	—
114	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
115	C₈	C₂×C₄⋊Dic₅₃	1	6	yes	—
116	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
117	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
118	C₂²	#16810449	1	16	yes	direct product factor
119	C₂²	C₂×#13133151	1	6	yes	—
120	C₂²	C₂×D₈₄₈	1	6	yes	—
121	C₂²	C₂×Dic₄₂₄	1	6	yes	—
122	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
123	C₄	—	424	8	no	—
124	C₄	C₂×#13133151	1	6	yes	—
125	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
126	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
127	C₂	C₂×#16810449	1	12	yes	direct product factor
128	C₂	C₂²×#13133151	1	1	yes	characteristic
129	C₂	C₂²×D₈₄₈	1	1	yes	characteristic
130	C₂	C₂²×Dic₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×#366
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#16810449	C₂	direct (N × H)
7. #16810449	C₂²	direct (N × H)
9. C₂²×C₁₆×C₅₃	C₄	semidirect
21. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
37. C₁₆×C₅₃	C₂²×C₄	semidirect
53. C₂²×C₅₃	#366	semidirect
65. C₂×C₅₃	C₂×#366	semidirect
71. C₂²×C₁₆	Dic₅₃	semidirect
76. C₅₃	C₂²×#366	semidirect
85. C₂×C₁₆	C₂×Dic₅₃	semidirect
103. C₁₆	C₂²×Dic₅₃	semidirect
118. C₂²	#16810449	direct (N × H)
127. C₂	C₂×#16810449	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133151	2
C₂×#16810449	2
C₂³×C₁₆×C₅₃	2
C₂²×#366	53

Subgroups — C22×#16810449 #27537113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810449 #27537113